Fix an \(I\) -graded vector space \({\mathbf {V}}=\bigoplus _{i\in I}{\mathbf {V}}_i\)  with dimension vector \(\nu \in \mathbb {N}I\)  and denote by \(E_{{\mathbf {V}}}\)  the variety consisting of representations of the quiver \(Q\) .
There is a \(G_{\mathbf {V}}=\prod _{i\in I}GL({\mathbf {V}}_i)\)  action on \(E_{\mathbf {V}}\) .
Denote by \({\mathcal {D}}_{G_{\mathbf {V}}}^b(E_{\mathbf {V}})\)  the \(G_{\mathbf {V}}\) -equivariant bounded derived category of \(\overline{{\mathbb {Q}}}_l\) -constructible mixed complexes on \(E_{\mathbf {V}}\) 
and by \({\mathcal {D}}_{G_{\mathbf {V}}}^{b,ss}(E_{\mathbf {V}})\)  the subcategory of \({\mathcal {D}}_{G_{\mathbf {V}}}^b(E_{\mathbf {V}})\)  consisting of semisimple complexes (see [1]}, [2]}).
